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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 203490.r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 203490.r1 | 203490db8 | \([1, -1, 0, -230432175, 1153753237611]\) | \(1856203306931677398202594801/285442925181016286066250\) | \(208087892456960872542296250\) | \([6]\) | \(71663616\) | \(3.7735\) | |
| 203490.r2 | 203490db6 | \([1, -1, 0, -221146605, 1265831924625]\) | \(1640729605302312040170582481/50078778067225044900\) | \(36507429211007057732100\) | \([2, 6]\) | \(35831808\) | \(3.4270\) | |
| 203490.r3 | 203490db3 | \([1, -1, 0, -221144985, 1265851396701]\) | \(1640693548282750959454626961/1528553391120\) | \(1114315422126480\) | \([6]\) | \(17915904\) | \(3.0804\) | |
| 203490.r4 | 203490db7 | \([1, -1, 0, -211886955, 1376664379335]\) | \(-1443141263044885978311580081/287846712789197778248970\) | \(-209840253623325180343499130\) | \([6]\) | \(71663616\) | \(3.7735\) | |
| 203490.r5 | 203490db5 | \([1, -1, 0, -61513425, -185498636139]\) | \(35310666410995026859894801/40072943900390625000\) | \(29213176103384765625000\) | \([2]\) | \(23887872\) | \(3.2242\) | |
| 203490.r6 | 203490db2 | \([1, -1, 0, -4836105, -1286010675]\) | \(17158661194925340654481/8947893637809000000\) | \(6523014461962761000000\) | \([2, 2]\) | \(11943936\) | \(2.8776\) | |
| 203490.r7 | 203490db1 | \([1, -1, 0, -2736585, 1728480141]\) | \(3109017019607132956561/30145442277888000\) | \(21976027420580352000\) | \([2]\) | \(5971968\) | \(2.5311\) | \(\Gamma_0(N)\)-optimal |
| 203490.r8 | 203490db4 | \([1, -1, 0, 18248895, -10025991675]\) | \(921946855702725447905519/593047570085451873000\) | \(-432331678592294415417000\) | \([2]\) | \(23887872\) | \(3.2242\) |
Rank
sage: E.rank()
The elliptic curves in class 203490.r have rank \(1\).
Complex multiplication
The elliptic curves in class 203490.r do not have complex multiplication.Modular form 203490.2.a.r
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 3 & 6 & 12 & 12 \\ 2 & 1 & 2 & 2 & 6 & 3 & 6 & 6 \\ 4 & 2 & 1 & 4 & 12 & 6 & 3 & 12 \\ 4 & 2 & 4 & 1 & 12 & 6 & 12 & 3 \\ 3 & 6 & 12 & 12 & 1 & 2 & 4 & 4 \\ 6 & 3 & 6 & 6 & 2 & 1 & 2 & 2 \\ 12 & 6 & 3 & 12 & 4 & 2 & 1 & 4 \\ 12 & 6 & 12 & 3 & 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
